ed to one, viz. that when the thing found by the rules in the first table is either a tangent or a cosine; and when, of the tangents or cosines employed in the computation of it, one only belongs to an obtuse angle, the angle required is also obtuse.

Thus, in the 15th case, when cos AB is found, if C be an obtuse angle, because of cos C, AB must be obtuse; and in case 16. if either B or C be obtuse, BC is greater than 90°, but if B and C are either both acute, or both obtuse, BC is less than 90°.

It is evident, that this rule does not apply when that which is found is the sine of an arch; and this, besides the three ambiguous cases, happens also in other two viz. the 1st and 11th. The ambiguity is obviated, in these two cases, by this rule, that the sides of a spherical right angled triangle are of the same affection with the opposite angles. Two rules are therefore sufficient to remove the ambiguity in all the cases of the right angled triangle, in which it can possibly be removed.

It may be useful to express the same solutions as in the annexed table. Let A be at the right angle as in the figure, and let the side opposite to it be a; let b be the side opposite to B, and c the side opposite to C.

PROBLEM II.

In any oblique angled spherical triangle, of the three sides and three angles, any three being given, it is required to find the other three.

In this Table, the references (c. 4.), (c. 5.), &c. are to the cases in the preceding Table, (16.), (27.), &c. to the propositions in Spherical Trigonometry.

AB, AC,

and the in

Let fall the perpendicular CD
from one of the unknown
angles on the side AB.

cluded angle The third R cos A:: tan AC: tan AD,

(c. 2.); therefore BD is
known,and cos AD : cos BD::
cos AC: cos BC, (26); ac-
cording as the segments AD
and DB are of the same or
different affection, AC and
CB will be of the same or
different affection.

Sin BC: sin AC :: sin A: sin B, (24). The affection of B is ambiguous, unless it can be determined by this rule, that according as AC + BC is greater or less than 180o, A + B is greater or less than 180o, (10).

From ACB the angle sought draw
CD perpendicular to AB; then
R: cos AC: tan A: cot ACD,
(c. 3.); and tan BC : tan AC ::
cos ACD: cos BCD, (28.)
ACDBCD = ACB, and
ACB is ambiguous, because of
the ambiguous sign + or -

Let fall the perpendicular CD from
the angle C, contained by the
given sides, upon the side AB
Rços A tan AC: tan AD,
(c. 2.); cos AC: cos BC:
cos AD: cos BD, (26.)
AB=AD ± BD, wherefore A
is ambiguous.